Properties

Label 950.2.a.h
Level $950$
Weight $2$
Character orbit 950.a
Self dual yes
Analytic conductor $7.586$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [950,2,Mod(1,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("950.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} + \beta q^{7} + q^{8} + (\beta + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} + \beta q^{7} + q^{8} + (\beta + 1) q^{9} + 4 q^{11} + \beta q^{12} + ( - 3 \beta + 2) q^{13} + \beta q^{14} + q^{16} + (\beta - 6) q^{17} + (\beta + 1) q^{18} - q^{19} + (\beta + 4) q^{21} + 4 q^{22} - 3 \beta q^{23} + \beta q^{24} + ( - 3 \beta + 2) q^{26} + ( - \beta + 4) q^{27} + \beta q^{28} + ( - 3 \beta + 2) q^{29} - 2 \beta q^{31} + q^{32} + 4 \beta q^{33} + (\beta - 6) q^{34} + (\beta + 1) q^{36} + 6 q^{37} - q^{38} + ( - \beta - 12) q^{39} + (4 \beta + 2) q^{41} + (\beta + 4) q^{42} + ( - 2 \beta + 8) q^{43} + 4 q^{44} - 3 \beta q^{46} + ( - 4 \beta + 4) q^{47} + \beta q^{48} + (\beta - 3) q^{49} + ( - 5 \beta + 4) q^{51} + ( - 3 \beta + 2) q^{52} + (\beta + 2) q^{53} + ( - \beta + 4) q^{54} + \beta q^{56} - \beta q^{57} + ( - 3 \beta + 2) q^{58} + \beta q^{59} + (2 \beta + 6) q^{61} - 2 \beta q^{62} + (2 \beta + 4) q^{63} + q^{64} + 4 \beta q^{66} + \beta q^{67} + (\beta - 6) q^{68} + ( - 3 \beta - 12) q^{69} + 4 \beta q^{71} + (\beta + 1) q^{72} + (3 \beta - 6) q^{73} + 6 q^{74} - q^{76} + 4 \beta q^{77} + ( - \beta - 12) q^{78} - 2 \beta q^{79} - 7 q^{81} + (4 \beta + 2) q^{82} + (2 \beta - 8) q^{83} + (\beta + 4) q^{84} + ( - 2 \beta + 8) q^{86} + ( - \beta - 12) q^{87} + 4 q^{88} + 2 q^{89} + ( - \beta - 12) q^{91} - 3 \beta q^{92} + ( - 2 \beta - 8) q^{93} + ( - 4 \beta + 4) q^{94} + \beta q^{96} - 6 q^{97} + (\beta - 3) q^{98} + (4 \beta + 4) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} + q^{6} + q^{7} + 2 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} + q^{6} + q^{7} + 2 q^{8} + 3 q^{9} + 8 q^{11} + q^{12} + q^{13} + q^{14} + 2 q^{16} - 11 q^{17} + 3 q^{18} - 2 q^{19} + 9 q^{21} + 8 q^{22} - 3 q^{23} + q^{24} + q^{26} + 7 q^{27} + q^{28} + q^{29} - 2 q^{31} + 2 q^{32} + 4 q^{33} - 11 q^{34} + 3 q^{36} + 12 q^{37} - 2 q^{38} - 25 q^{39} + 8 q^{41} + 9 q^{42} + 14 q^{43} + 8 q^{44} - 3 q^{46} + 4 q^{47} + q^{48} - 5 q^{49} + 3 q^{51} + q^{52} + 5 q^{53} + 7 q^{54} + q^{56} - q^{57} + q^{58} + q^{59} + 14 q^{61} - 2 q^{62} + 10 q^{63} + 2 q^{64} + 4 q^{66} + q^{67} - 11 q^{68} - 27 q^{69} + 4 q^{71} + 3 q^{72} - 9 q^{73} + 12 q^{74} - 2 q^{76} + 4 q^{77} - 25 q^{78} - 2 q^{79} - 14 q^{81} + 8 q^{82} - 14 q^{83} + 9 q^{84} + 14 q^{86} - 25 q^{87} + 8 q^{88} + 4 q^{89} - 25 q^{91} - 3 q^{92} - 18 q^{93} + 4 q^{94} + q^{96} - 12 q^{97} - 5 q^{98} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.56155
2.56155
1.00000 −1.56155 1.00000 0 −1.56155 −1.56155 1.00000 −0.561553 0
1.2 1.00000 2.56155 1.00000 0 2.56155 2.56155 1.00000 3.56155 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( +1 \)
\(19\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.a.h 2
3.b odd 2 1 8550.2.a.br 2
4.b odd 2 1 7600.2.a.y 2
5.b even 2 1 190.2.a.d 2
5.c odd 4 2 950.2.b.f 4
15.d odd 2 1 1710.2.a.w 2
20.d odd 2 1 1520.2.a.n 2
35.c odd 2 1 9310.2.a.bc 2
40.e odd 2 1 6080.2.a.bb 2
40.f even 2 1 6080.2.a.bh 2
95.d odd 2 1 3610.2.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.d 2 5.b even 2 1
950.2.a.h 2 1.a even 1 1 trivial
950.2.b.f 4 5.c odd 4 2
1520.2.a.n 2 20.d odd 2 1
1710.2.a.w 2 15.d odd 2 1
3610.2.a.t 2 95.d odd 2 1
6080.2.a.bb 2 40.e odd 2 1
6080.2.a.bh 2 40.f even 2 1
7600.2.a.y 2 4.b odd 2 1
8550.2.a.br 2 3.b odd 2 1
9310.2.a.bc 2 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(950))\):

\( T_{3}^{2} - T_{3} - 4 \) Copy content Toggle raw display
\( T_{7}^{2} - T_{7} - 4 \) Copy content Toggle raw display
\( T_{11} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$11$ \( (T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - T - 38 \) Copy content Toggle raw display
$17$ \( T^{2} + 11T + 26 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 3T - 36 \) Copy content Toggle raw display
$29$ \( T^{2} - T - 38 \) Copy content Toggle raw display
$31$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$37$ \( (T - 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 8T - 52 \) Copy content Toggle raw display
$43$ \( T^{2} - 14T + 32 \) Copy content Toggle raw display
$47$ \( T^{2} - 4T - 64 \) Copy content Toggle raw display
$53$ \( T^{2} - 5T + 2 \) Copy content Toggle raw display
$59$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$61$ \( T^{2} - 14T + 32 \) Copy content Toggle raw display
$67$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$71$ \( T^{2} - 4T - 64 \) Copy content Toggle raw display
$73$ \( T^{2} + 9T - 18 \) Copy content Toggle raw display
$79$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$83$ \( T^{2} + 14T + 32 \) Copy content Toggle raw display
$89$ \( (T - 2)^{2} \) Copy content Toggle raw display
$97$ \( (T + 6)^{2} \) Copy content Toggle raw display
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